Quote Originally Posted by Panconleche
Hello,
I am taking the first part of Topology and I am having a little trouble understanding the concept of a Compact Set. The biggest problem to me is the definition of Compact, open cover, and sub cover. I have read the material and try to make sense of it, however, I am still a little lost with the material. If you could shine some light my way I would appreciate it.

Thank you
A subset H of a Euclidean space X is said to be compact (in X) if every open covering of H has a finite subcovering.

Here are a few examples.

Consider the interval [1,2] can and the cover \{ 1-\frac{1}{n},2+\frac{1}{n}\}_{n \in \mathbb{N}}

This covers [1,2] for all values of n but it also covers for any finite value of n.

Now consider (1,2) and the cover above this covers it, but it must be true for every covering.

Now consider (1+\frac{1}{n},2-\frac{1}{n})_{n \in \mathbb{N}}
at "infinity" it covers (1,2) but doesn't for any finite value of n.
so (1,2) is not compact

I hope this helps.

Brett